(488a) Economic Nonlinear Model Predictive Control of an Integrated Solid-Sorbent Carbon Capture System
The major components of the solid-based carbon capture system are bubbling fluidized bed (BFB) adsorber and regenerator, which are described by a one-dimensional, three-region, non-isothermal and spatially distributed model. The reactorâ??s hydrodynamic behavior is described by partial differential and algebraic equations (PDAEs), constructed from mass and heat conservation and hydrodynamic correlations. The complete description of the BFB model can be found in [1, 2]. After spatial discretization, the BFB model becomes a highly nonlinear and large scale differential and algebraic equation (DAE) system which consists of 2000 differential equations and over 10000 algebraic equations. It is difficult to directly apply this rigorous model for time-critical applications such as NMPC. Therefore, we developed dynamic reduced models to reduce computational cost while maintaining accurate prediction capacity of the rigorous model.
Physics-based model reduction of the rigorous BFB model was conducted for both spatial and temporal aspects . Orthogonal collocation on finite elements (OCFE) was applied to discretize the partial differential equations in space. Since OCFE is high order, it requires fewer discretization points to achieve the accuracy of the finite difference method. Further model reduction is due to unevenly distributed finite elements in space. For the temporal reduction, eigenvalue analysis was used to assess overall time scale differences in the rigorous BFB model. Then a quasi-steady state approximation replaced differential equations with algebraic states. In addition, a null space projection method eliminated reversible reactions for CO2 adsorption and led to a simplified kinetic model. The resulting reduced system therefore has essentially the same general behavior, but becomes much less stiff.
To maintain high CO2 capture under unsteady operation we develop a dynamic optimization strategy that discretizes the differential-algebraic equations in time and solves the corresponding nonlinear programming (NLP) problem. Moreover, we extend this approach to on-line optimization through NMPC. For a sufficiently long predictive horizon, the corresponding NLP problem raises a computational challenge, which we address through input and state blocking which significantly reduces the size of NLP subproblems for economic NMPC and maintains Lyapunov stability . Based on the process dynamics, these nonuniform discretizations for state and control profiles also capture multiple time scales of the system and lead to control performance that is essentially the same as using uniform grids.
In this talk, we compare this approach with traditional setpoint-tracking NMPC, and we demonstrate that economic NMPC achieves significant reduction in the operational cost of the integrated carbon capture system, by directly considering process economics in the objective function. With reduced model and blocking strategies, we obtain a much smaller optimization problem and achieve one order of magnitude reduction in the solution time for economic NMPC problems. Moreover, advanced step NMPC (asNMPC)  is then applied to take these optimization calculations off-line in order to enable fast online control of the integrated carbon capture system. In addition, we apply the robust problem reformulation in  and relax inequality constraints with l1 penalty terms, with sufficiently large penalty parameters, to guarantee stability and improve NMPC robustness under process disturbance/uncertainty.
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 Modekurti, Srinivasarao, Debangsu Bhattacharyya, and Stephen E. Zitney. "Dynamic Modeling and Control Studies of a Two-Stage Bubbling Fluidized Bed Adsorber-Reactor for Solidâ??Sorbent CO2 Capture."Â Industrial & Engineering Chemistry Research 52.30 (2013): 10250-10260.
 Yu, Mingzhao, David C. Miller, and Lorenz T. Biegler. "Dynamic reduced order models for simulating bubbling fluidized bed adsorbers." Industrial & Engineering Chemistry Research 54.27 (2015): 6959-6974.
 Yu, Mingzhao and Lorenz T. Biegler. A Stable and Robust NMPC Strategy with Reduced Models and Nonuniform Grids, the 11th IFAC Symposium on Dynamics and Control of Process Systems, 2016
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 Yang, Xue, Devin W. Griffith, and Lorenz T. Biegler. "Nonlinear Programming Properties for Stable and Robust NMPC." IFAC-PapersOnLine 48.23 (2015): 388-397.